This question was inspired by a section in the 3blue1brown video on fractal dimension, where it is described how the box-counting dimension can change based on zoom level.
I have no idea which topic this question comes under. Feel free to retag with appropriate tags.
I was thinking about a shape where the local/apparent dimension changes with how much we zoom into the structure.
Imagine a helix embedded in $3$D space. It appears $1$D at a large scale, but when we zoom into, we find that the “wireframe” of the helix is actually a tube-like surface, thus making it appear a $2$D shape. Now, when we zoom in ever further on the tubing, we find that the tube is actually a helix with very closely-spaced coils, and sufficient zooming makes it appear $1$D again.
Have such shapes, where the apparent dimension changes based on the zoom level, been studied in previous mathematical literature? Is there any particular “keyword” to refer to such shapes?
